Probability of a continuous function to be differentiable

Question

Assume I have a little bit of knowledge in topology and I want to prove, in a full formal way, that the probability of a continuous function, say on the interval $ [0,1]$ to be differentiable, say in at least one dot, is zero.

I guess what would be require is the following:

Prove that the set of all function on $ [0,1]$ which is differentiable at least in one point, is a meager set, or first category Baire set. (I am familier with Baire Category Theorem so if this is an easy consequence I'd be happy to see).

2.Define some suitable probability measure on the space of continuous functions on $ [0,1]$ .

3.Prove that first categroy Baire sets happen to occur with probability 0.

Is there a published proof somewhere? Is it easy to prove following the steps I mentioned?

Thanks in advance.

A question about probability is only meaningful when you have specified the probability space you are working in: so you must carry out your item 2 to make your question definite before deciding how you might answer it. — Rob Arthan, Jan 15 '21 at 22:01
Typically Baire category 1 sets typically don't have measure $0$. (connected to the problem of category measures, which are rare). $[0,1]$ has a Baire first category set of measure 1, e.g. — Henno Brandsma, Jan 15 '21 at 22:36
@RobArthan When people say that the probability of a continuous function to be differentiable is zero, what is the probability space that they are talking about? — FreeZe, Jan 16 '21 at 01:47
Does this answer your question? Are "most" continuous functions also differentiable? — Peter O., Nov 13 '21 at 19:49

Probability of a continuous function to be differentiable

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